Group #25017392 — subgroups

Subgroups — #25017392

Order 11008.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582739	C₂	1	2	yes	maximal
2	#19136848	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₃⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#19132775	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₄	—	2	1	no	—
7	C₂×Dic₃₄₄	C₂²	1	2	yes	—
8	#14583279	—	2	1	no	—
9	#14582739	—	2	4	no	—
10	C₂×C₄₃⋊C₃₂	C₂²	1	2	yes	—
11	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic
12	C₄×Q₁₆×C₄₃	—	2	1	no	—
13	C₄₃×#366	C₂²	1	1	yes	characteristic
14	C₄₃×#358	—	2	1	no	—
15	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₂	—	2	2	no	—
17	C₂×Dic₁₇₂	—	4	1	no	—
18	#13063325	—	4	1	no	—
19	Dic₃₄₄	—	2	2	no	—
20	Dic₃₄₄	—	2	2	no	—
21	Dic₃₄₄	—	4	2	no	—
22	C₄₃⋊C₃₂	—	2	2	no	—
23	C₂×Q₁₆×C₄₃	—	2	1	no	—
24	C₄×Q₈×C₄₃	—	4	1	no	—
25	Q₃₂×C₄₃	—	2	1	no	—
26	Q₃₂×C₄₃	D₄	1	2	yes	—
27	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
29	#107×C₄₃	—	2	1	no	—
30	#103×C₄₃	—	4	1	no	—
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₂×Q₈×C₄₃	—	4	1	no	—
33	C₂×Dic₈₆	—	4	2	no	—
34	Q₁₆×C₄₃	—	2	2	no	—
35	Q₁₆×C₄₃	—	4	1	no	—
36	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₄₃	—	4	1	no	—
39	C₄⋊C₄×C₄₃	—	8	1	no	—
40	C₄²×C₄₃	—	8	1	no	—
41	C₄²×C₄₃	D₈	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	8	1	no	—
43	C₄⋊Dic₄₃	—	8	1	no	—
44	Dic₁₇₂	—	4	2	no	—
45	Dic₁₇₂	—	4	2	no	—
46	Dic₁₇₂	—	8	2	no	—
47	C₁₆×C₄₃	—	2	1	no	—
48	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₄₃	#41	1	1	yes	characteristic
50	Q₈×C₄₃	—	4	2	no	—
51	Q₈×C₄₃	—	8	1	no	—
52	C₂×C₄×C₄₃	—	8	1	no	—
53	C₂×C₄×C₄₃	—	8	1	no	—
54	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
57	C₂×Dic₄₃	—	8	2	no	—
58	Dic₈₆	—	8	2	no	—
59	Dic₈₆	—	8	2	no	—
60	C₈×C₄₃	—	2	1	no	—
61	C₈×C₄₃	#121	1	1	yes	characteristic
62	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	#13688	—	43	1	no	maximal, Sylow
64	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	—	8	2	no	—
67	C₄×C₄₃	—	16	1	no	—
68	C₄×C₄₃	Q₆₄	1	2	yes	complemented
69	C₄×C₄₃	#459	1	1	yes	characteristic
70	C₄×C₄₃	#509	1	1	yes	characteristic
71	Dic₄₃	—	16	2	no	—
72	C₂×Q₆₄	—	43	2	no	—
73	#2247	—	43	1	no	—
74	C₄×Q₃₂	D₄₃	1	1	yes	characteristic
75	C₄⋊C₃₂	—	43	1	no	—
76	#1416	—	43	2	no	—
77	C₂×C₄₃	#2206	1	1	yes	characteristic
78	C₂×C₄₃	C₂×Q₆₄	1	1	yes	characteristic
79	C₂×C₄₃	#2264	1	1	yes	characteristic
80	C₂×Q₃₂	—	43	2	no	—
81	C₂×Q₃₂	—	86	1	no	—
82	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic
83	#494	—	86	1	no	—
84	C₄×Q₁₆	—	2	1	no	—
85	Q₆₄	—	86	4	no	—
86	C₂×C₃₂	—	43	2	no	—
87	#366	C₂×D₄₃	1	1	yes	characteristic
88	#358	—	2	1	no	—
89	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
90	C₄₃	#13688	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	86	2	no	—
93	C₂×Q₁₆	—	172	1	no	—
94	C₄⋊Q₈	—	172	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	Q₃₂	—	2	1	no	—
97	Q₃₂	—	86	2	no	—
98	Q₃₂	—	86	2	no	—
99	Q₃₂	—	172	2	no	—
100	Q₃₂	D₁₇₂	1	2	yes	—
101	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
103	#107	—	2	1	no	—
104	#103	—	4	1	no	—
105	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
106	C₃₂	—	86	2	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	172	2	no	—
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	4	1	no	—
111	Q₁₆	—	344	2	no	—
112	Q₁₆	—	172	2	no	—
113	Q₁₆	—	172	2	no	—
114	C₂×C₈	#10621882	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	344	1	no	—
119	C₄⋊C₄	—	344	1	no	—
120	C₄²	—	8	1	no	—
121	C₄²	#10621882	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	C₂×D₁₇₂	1	1	yes	characteristic
124	C₁₆	#10621898	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	Q₈	—	344	2	no	—
128	Q₈	—	344	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	344	2	no	—
132	C₂×C₄	#13063266	1	1	yes	characteristic
133	C₂×C₄	#13063266	1	1	yes	characteristic
134	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#13063299	1	1	yes	characteristic
137	C₈	#13063350	1	1	yes	characteristic
138	C₂²	C₂×#13063266	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	688	2	no	—
143	C₄	#14582739	1	2	yes	complemented
144	C₄	#14583094	1	1	yes	characteristic
145	C₄	#14583334	1	1	yes	characteristic
146	C₂	#19136630	1	1	yes	characteristic
147	C₂	C₂×#14582739	1	1	yes	characteristic
148	C₂	#19136912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582739

C₂

yes

maximal

#19136848

C₂

yes

maximal, characteristic

C₄⋊(C₄₃⋊C₃₂)

C₂

yes

maximal, characteristic

#19132775

C₂

yes

maximal

C₄×Q₃₂×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

—

C₂×Dic₃₄₄

C₂²

yes

—

#14583279

—

#14582739

—

C₂×C₄₃⋊C₃₂

C₂²

yes

—

C₂×Q₃₂×C₄₃

C₂²

yes

characteristic

C₄×Q₁₆×C₄₃

—

C₄₃×#366

C₂²

yes

characteristic

C₄₃×#358

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

Dic₃₄₄

—

Dic₃₄₄

—

Dic₃₄₄

—

C₄₃⋊C₃₂

—

C₂×Q₁₆×C₄₃

—

C₄×Q₈×C₄₃

—

Q₃₂×C₄₃

—

Q₃₂×C₄₃

D₄

yes

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

#107×C₄₃

—

#103×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₈

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×D₄

yes

characteristic

C₁₆×C₄₃

#41

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

yes

C₈×C₄₃

yes

—

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

Q₆₄

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

D₄₃

yes

characteristic

C₄⋊C₃₂

—

#1416

—

C₂×C₄₃

#2206

yes

characteristic

C₂×C₄₃

C₂×Q₆₄

yes

characteristic

C₂×C₄₃

#2264

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₄₃

yes

characteristic

#494

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

—

#366

C₂×D₄₃

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄₃

#13688

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

172

—

C₄⋊Q₈

—

172

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

172

—

100

Q₃₂

D₁₇₂

yes

—

101

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

102

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

103

#107

—

104

#103

—

105

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

106

C₃₂

—

107

C₂×Q₈

—

108

C₂×Q₈

—

172

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

344

—

112

Q₁₆

—

172

—

113

Q₁₆

—

172

—

114

C₂×C₈

#10621882

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

344

—

119

C₄⋊C₄

—

344

—

120

C₄²

—

121

C₄²

#10621882

yes

characteristic

122

C₁₆

—

123

C₁₆

C₂×D₁₇₂

yes

characteristic

124

C₁₆

#10621898

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

344

—

128

Q₈

—

344

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

344

—

132

C₂×C₄

#13063266

yes

characteristic

133

C₂×C₄

#13063266

yes

characteristic

134

C₂×C₄

C₂×#10621882

yes

characteristic

135

C₈

—

136

C₈

#13063299

yes

characteristic

137

C₈

#13063350

yes

characteristic

138

C₂²

C₂×#13063266

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

688

—

143

C₄

#14582739

yes

complemented

144

C₄

#14583094

yes

characteristic

145

C₄

#14583334

yes

characteristic

146

C₂

#19136630

yes

characteristic

147

C₂

C₂×#14582739

yes

characteristic

148

C₂

#19136912

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13063266
commutator, cosocle	C₂×C₁₆×C₄₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄×Q₃₂×C₄₃	C₂
socle, prime radical	C₂²×C₄₃	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₄₃
hypocenter	C₄₃	#13688
LCS	G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#13063266

commutator, cosocle

C₂×C₁₆×C₄₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄×Q₃₂×C₄₃

C₂

socle, prime radical

C₂²×C₄₃

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₄₃

hypocenter

C₄₃

#13688

LCS

G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
68. C₄×C₄₃	Q₆₄	semidirect
90. C₄₃	#13688	semidirect
143. C₄	#14582739	semidirect

H ≅ G/N

type

68. C₄×C₄₃

Q₆₄

semidirect

90. C₄₃

#13688

semidirect

143. C₄

#14582739

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14582739	2
#19136848	2
C₄⋊(C₄₃⋊C₃₂)	2
#19132775	2
C₄×Q₃₂×C₄₃	2
#13688	43

subgroup

index

C₄⋊(C₄₃⋊C₃₂)

C₄×Q₃₂×C₄₃

Subgroups — #25017392

Conjugacy classes of subgroups

Complements

Maximal subgroups